Results 1  10
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176
Algebras and Modules in Monoidal Model Categories
 Proc. London Math. Soc
, 1998
"... In recent years the theory of structured ring spectra (formerly known as A #  and E # ring spectra) has been signicantly simplified by the discovery of categories of spectra with strictly associative and commutative smash products. Now a ring spectrum can simply be dened as a monoid with respect t ..."
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Cited by 143 (27 self)
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In recent years the theory of structured ring spectra (formerly known as A #  and E # ring spectra) has been signicantly simplified by the discovery of categories of spectra with strictly associative and commutative smash products. Now a ring spectrum can simply be dened as a monoid with respect to the smash product in one of these new categories of spectra. In order to make use of all of the standard tools from homotopy theory, it is important to have a Quillen model category structure [##] available here. In this paper we provide a general method for lifting model structures to categories of rings, algebras, and modules. This includes, but is not limited to, each of the new theories of ring spectra. One model for structured ring spectra is given by the Salgebras of [##]. This example has the special feature that every object is brant, which makes it easier to fo...
Model categories of diagram spectra
 Proc. London Math. Soc
"... 1. Preliminaries about topological model categories 5 2. Preliminaries about equivalences of model categories 9 3. The level model structure on Dspaces 10 4. Preliminaries about π∗isomorphisms of prespectra 14 ..."
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Cited by 111 (35 self)
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1. Preliminaries about topological model categories 5 2. Preliminaries about equivalences of model categories 9 3. The level model structure on Dspaces 10 4. Preliminaries about π∗isomorphisms of prespectra 14
Stable model categories are categories of modules
 TOPOLOGY
, 2003
"... A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for ..."
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Cited by 78 (16 self)
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A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard’s work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the EilenbergMac Lane spectrum HR and (unbounded) chain complexes of Rmodules for a ring R.
Presheaves of symmetric spectra
, 1998
"... The main theorem of [4] say that there is a proper closed simplicial model category ..."
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Cited by 75 (12 self)
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The main theorem of [4] say that there is a proper closed simplicial model category
On differential graded categories
 INTERNATIONAL CONGRESS OF MATHEMATICIANS. VOL. II
, 2006
"... Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, DuggerShipley,..., Toën and ToënVaquié. ..."
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Cited by 63 (3 self)
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Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, DuggerShipley,..., Toën and ToënVaquié.
Equivariant orthogonal spectra and Smodules
 MR1922205 (2003i:55012), Zbl 1025.55002
"... 1. Right exact functors on categories of diagram spaces 2 2. The proofs of the comparison theorems 5 3. The construction of the functor N ∗ 12 ..."
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Cited by 61 (8 self)
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1. Right exact functors on categories of diagram spaces 2 2. The proofs of the comparison theorems 5 3. The construction of the functor N ∗ 12
Spectra and symmetric spectra in general model categories
 J. Pure Appl. Algebra
"... Abstract. We give two general constructions for the passage from unstable to stable homotopy that apply to the known example of topological spaces, but also to new situations, such as the A1homotopy theory of MorelVoevodsky [16, 23]. One is based on the standard notion of spectra originated by Boa ..."
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Cited by 55 (0 self)
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Abstract. We give two general constructions for the passage from unstable to stable homotopy that apply to the known example of topological spaces, but also to new situations, such as the A1homotopy theory of MorelVoevodsky [16, 23]. One is based on the standard notion of spectra originated by Boardman [24]. Its input is a wellbehaved model category C and an endofunctor
Ideals in triangulated categories: Phantoms, ghosts and skeleta
 Adv. in Math
, 1998
"... ABSTRACT. We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal I of morphisms with certain properties, and that if I has these properties, then so does each of its powers. We show how a projective class leads to an Adams spectral seq ..."
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Cited by 41 (5 self)
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ABSTRACT. We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal I of morphisms with certain properties, and that if I has these properties, then so does each of its powers. We show how a projective class leads to an Adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence. We use this to study various ideals. In the stable homotopy category we examine phantom maps, skeletal phantom maps, superphantom maps, and ghosts. (A ghost is a map which induces the zero map of homotopy groups.) We show that ghosts lead to a stable analogue of the Lusternik–Schnirelmann category of a space, and we calculate this stable analogue for lowdimensional real projective spaces. We also give a relation between ghosts and the Hopf and Kervaire invariant problems. In the case of A ∞ modules over an A ∞ ring spectrum, the ghost spectral sequence is a universal coefficient spectral sequence. From the phantom projective class we derive a generalized Milnor sequence for filtered diagrams of finite spectra, and from this it follows that the group of phantom maps from X to Y can always be described as a lim1 ←− group. The last two sections focus
A convenient model category for commutative ring spectra
, 2003
"... We develop a new system of model structures on the modules, algebras and commutative algebras over symmetric spectra. In addition to the same properties as the standard stable model structures defined in [HSS] and [MMSS], these model structures have better compatibility properties between commutati ..."
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Cited by 39 (2 self)
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We develop a new system of model structures on the modules, algebras and commutative algebras over symmetric spectra. In addition to the same properties as the standard stable model structures defined in [HSS] and [MMSS], these model structures have better compatibility properties between commutative algebras and the underlying modules.
Homotopical Algebraic Geometry I: Topos theory
, 2002
"... This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ..."
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Cited by 32 (20 self)
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This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ∞categories, and we develop the notions of Stopologies, Ssites and stacks over them. We prove in particular, that for an Scategory T endowed with an Stopology, there exists a model